In this Section, we apply our perturbation expansion to a physically relevant system: soft disks with small polydispersity in particle sizes.
We begin with a triangular crystal composed of deformable disks interacting via one sided pairwise potentials that is now paradigmatic in the study of soft particles and deformable foams [1]}, [2]}. The form of the potential is
\(\nonumber V_{\sigma _{ij}}(\vec{r}_{ij}) &=& \frac{K}{\alpha }\left(1- \frac{| \vec{r}_{ij}|}{\sigma _{ij}}\right)^\alpha ~~\textmd {for}~~ r_{ij} < \sigma _{ij},\\&=& 0 ~~~~~~~~~~~~~~~~~~~~~~\textmd {for}~~r_{ij} > \sigma _{ij}.\) 
